In the philosophy of mathematics, ultrafinitism, ultraintuitionism, strict formalism, strict finitism, and strong finitism

Lavine has shown that the basic principles of arithmetic such as "there is no largest natural number" can be upheld, as Lavine allows for the inclusion of "indefinitely large" numbers. and Vladimir Sazonov's notion of feasible numbers.<!-- seems unclear whether it is really *his* concept, someone can read attached to learn more. https://link.springer.com/chapter/10.1007/3-540-60178-3_78 -->

There has also been considerable formal development on versions of ultrafinitism that are based on complexity theory, like Samuel Buss's bounded arithmetic theories, which capture mathematics associated with various complexity classes like P and PSPACE. Buss's work can be considered the continuation of Edward Nelson's work on predicative arithmetic as bounded arithmetic theories like S12 are interpretable in Raphael Robinson's theory Q and therefore are predicative in Nelson's sense. The power of these theories for developing mathematics is studied in bounded reverse mathematics as can be found in the works of Stephen A. Cook and Phuong The Nguyen. However these are not philosophies of mathematics but rather the study of restricted forms of reasoning similar to reverse mathematics.

See also

  • Finitism
  • Internal set theory — An enrichment of ZFC which has theorems such as "there exists a largest standard natural number". Developed by ultrafinitist Edward Nelson.
  • Transcomputational problem

Notes

References

  • Reviewed by
  • Lavine, S., 1994. Understanding the Infinite, Cambridge, MA: Harvard University Press.
  • Explicit finitism by András Kornai
  • On feasible numbers by Vladimir Sazonov
  • "Real" Analysis Is A Degenerate Case Of Discrete Analysis by Doron Zeilberger
  • Discussion on formal foundations on MathOverflow
  • History of constructivism in the 20th century by A. S. Troelstra
  • Predicative Arithmetic by Edward Nelson
  • Logical Foundations of Proof Complexity by Stephen A. Cook and Phuong The Nguyen
  • Bounded Reverse Mathematics by Phuong The Nguyen
  • Reading Brian Rotman’s “Ad Infinitum…” by Charles Petzold
  • Computational Complexity Theory